a Xi :0910.1246 1 [ma h.AP] 7 Oc 2009
Heisenbe g uniqueness pai s and he Klein-Go don
equa ion
Håkan Hedenmalm and Al onso Mon es-Rod íguez
Abs ac . A Heisenbe g uniqueness pai (HUP) is a pai (Γ,Λ), whe e Γis a cu e in he plane
and Λis a se in he plane, wi h he ollowing p ope y: any bounded Bo el measu e µin he plane
suppo ed on Γ, which is absolu ely con inuous wi h espec o a c leng h, and whose Fou ie
ans o m b
µ anishes on Λ, mus au oma ically be he ze o measu e. We p o e ha when Γis he
hype bola x1x2=1, and Λis he la ice-c oss
Λ = (αZ×{0})∪({0}×βZ),
whe e α, β a e posi i e eals, hen (Γ,Λ) is an HUP i and only i αβ ≤1; in his si ua ion, he
Fou ie ans o m b
µo he measu e sol es he one-dimensional Klein-Go don equa ion. Ph ased
di e en ly, we show ha
eπiαn ,eπiβn/ ,n∈Z,
span a weak-s a dense subspace in L∞(R) i and only i αβ ≤1. In o de o p o e his heo em,
some elemen s o linea ac ional heo y and e godic heo y a e needed, such as he Bi kho
E godic Theo em. An idea pa allel o he one exploi ed by Maka o and Pol o a ski (in he
con ex o model subspaces) is also needed. As a consequence, we sol e a p oblem on he densi y
o algeb as gene a ed by wo inne unc ions aised by Ma heson and S essin.
1. In oduc ion
Heisenbe g uniqueness pai s. Le µbe a ini e complex- alued Bo el measu e in he plane R2,
and associa e o i he Fou ie ans o m
b
µ(ξ)=ZR2
eπihx,ξidµ(x),
whe e x=(x1,x2) and ξ=(ξ1, ξ2), wi h inne p oduc
hx, ξi=x1ξ1+x2ξ2.
The Heisenbe g unce ain y p inciple s a es ha bo h µand b
µcanno bo h be oo concen a ed
o a poin (see [6] o he o iginal pape o Heisenbe g, and [5] o a mo e gene al ea men );
in pa icula , hey canno bo h ha e compac suppo . He e, we shall s udy a a ia ion on ha
heme. Le Γbe a smoo h cu e in R2, o , mo e gene ally, a ini e disjoin union o smoo h cu es.
1991 Ma hema ics Subjec Classi ica ion. P ima y 42B10, 42A10, 58F11; Seconda y 11K50, 31B35, 43A15, 81Q05.
Key wo ds and ph ases. T igonome ic sys em, in e sion, composi ion ope a o , Klein-Go don equa ion, e godic heo y.
Resea ch pa ially suppo ed by he Gö an Gus a sson Founda ion and by he Swedish Science Council (Ve enskaps åde ).
Resea ch pa ially suppo ed by Plan Nacional I+D+I g an no. MTM2006-09060 and by Jun a de Andalucía g an s
nos. FQM-260 and FQM06-02225.
2 Hedenmalm and Mon es
Suppose ha supp µ⊂Γ, and ha µis absolu ely con inuous wi h espec o a c leng h measu e
on Γ. Which se s Λ⊂R2ha e he p ope y ha
b
µ|Λ=0=⇒µ=0?
I his is he case, we say ha (Γ,Λ) is a Heisenbe g uniqueness pai . A dual o mula ion is ha (Γ,Λ)
is a Heisenbe g uniqueness pai i and only i he unc ions
eξ(x)=eπihx,ξi, ξ ∈Λ,
span a weak-s a dense subspace in L∞(Γ). This concep o Heisenbe g uniqueness pai s has many
ea u es in common wi h he no ion o (weakly) mu ually annihila ing pai s o Bo el measu able
se s ha ing posi i e a ea measu e, which appea s, o ins ance, in he book by Ha in and Jö icke
[5].
The p ope ies o he Fou ie ans o m wi h espec o ansla ion and mul iplica ion by
complex exponen ials show ha o all poin s x∗, ξ∗∈R2, we ha e
(in -1) (Γ + {x∗},Λ + {ξ∗}) is an HUP ⇐⇒ (Γ,Λ) is an HUP,
whe e HUP is sho o “Heisenbe g uniqueness pai ”. Likewise, i is also s aigh o wa d o see
ha i T:R2→R2is an in e ible linea ans o ma ion wi h adjoin T∗, hen
(in -2) (T−1(Γ),T∗(Λ)) is an HUP ⇐⇒ (Γ,Λ) is an HUP.
Algeb aic cu es and pa ial di e en ial equa ions. Algeb aic cu es Γa e o pa icula in e es ,
because o hei connec ion o pa ial di e en ial equa ions. Tha connec ion ollows om he
obse a ion ha o polynomials po wo a iables,
p∂1
πi,∂2
πib
µ(ξ)=ZR2
eπihx,ξip(x1,x2) dµ(x),
so ha i pis eal- alued and Γis he locus o he equa ion
p(x1,x2)=0,
hen
p(x1,x2) dµ(x1,x2)=0
iden ically. The e o e b
µsol es he PDE
(1.1) p∂1
πi,∂2
πib
µ(ξ)=0
in he plane. In ac , he equa ion (1.1) encodes he equi emen ha supp µ⊂Γ.
Conic sec ions. We shall conside he case when Γis a conic sec ion, ha is, he locus o a quad a ic
equa ion
ax2
1+bx2
2+cx1x2+dx1+ex2+ =0,
whe e a,b,c,d,e, a e eal cons an s. As we only conside he case when Γis a cu e, his lea es us
wi h he ollowing cases: a s aigh line, wo pa allel s aigh lines, a c oss, an ellipse, a pa abola,
o a hype bola.
The line. Le us look a he line i s , as a model example. By he in a iance p ope ies (in -1) and
(in -2), we may assume ha Γ = R× {0}, he x1-axis. In his case, b
µ(ξ) depends only on ξ1, and
i is easy o see ha (Γ,Λ) is a Heisenbe g uniqueness pai i and only i π1(Λ), he o hogonal
p ojec ion o Λ o he ξ1-axis, is dense.
Two pa allel lines. I Γis he union o wo pa allel lines, we may wi hou loss o gene ali y assume
ha
Γ = R×{0,1}.
Heisenbe g uniqueness pai s and he Klein-Go don equa ion 3
In his case, we see om he example o he line ha i is necessa y o (Γ,Λ) o be a Heisenbe g
uniqueness pai ha π1(Λ) be dense. Bu some hing mo e is needed. An absolu ely con inuous
measu e µon Γmay be w i en in he o m
dµ(x)= (x1)dx1dδ0(x2)+g(x1)dx1dδ1(x2),
whe e ,g∈L1(R) (δydeno es he uni poin mass a he poin y), so ha
b
µ(ξ)=b
(ξ1)+eπiξ2b
g(ξ1).
Nex , we spli
π1(Λ)=πa
1(Λ)∪πb
1(Λ),
whe e he wo se s a e disjoin : ∈πa
1(Λ) i he e a e wo li ed poin s ξ=(ξ1, ξ2) and η=(η1, η2)
in Λ, wi h ξ1=η1= and ξ2−η2<2Z, whe eas ∈πb
1(Λ) i he la e does no happen. We quickly
ind ha
(1.2) b
( )=b
g( )=0, ∈πa
1(Λ).
On he o he hand, o ∈πb
1(Λ), he exp ession eπiξ2is a well-de ined unc ion o ξ1= , whe e ξ2
s ands o any o he poin s wi h (ξ1, ξ2)∈Λ; we w i e χ( ) o his unimodula unc ion. I Eis a
closed subse o Rand 0∈E, we say ha a unc ion ϕ:E→Cis locally he Fou ie ans o m o an
L1(R) unc ion a ound 0p o ided ha he e exis s a small open in e al Ia ound 0and a unc ion
ψwhich is he Fou ie ans o m o an L1(R) unc ion, such ha ψ=ϕon E∩I. Le πc
1(Λ) consis o
hose poin s 0∈πb
1(Λ) whe e χ:πb
1(Λ)→Cis locally he Fou ie ans o m o an L1(R) unc ion
a ound 0.
Theo em 1.1. (Γ,Λ)is a Heisenbe g uniqueness pai i and only i πa
1(Λ)∪(πb
1(Λ) πc
1(Λ)) is dense in R.
P oo . We obse e ha
(1.3) b
( )=−χ( )b
g( ), ∈πb
1(Λ).
I ∈πb
1(Λ) πc
1(Λ), his is only possible i b
g( )=0, so ha
(1.4) b
( )=b
g( )=0, ∈πb
1(Λ) πc
1(Λ).
A combina ion o (1.2) and (1.4) shows ha =g=0 (so ha µ=0) i he se πa
1(Λ)∪(πb
1(Λ) πc
1(Λ))
is dense in R.
As o he o he di ec ion, suppose ha π1(Λ) is dense in R, while πa
1(Λ)∪(πb
1(Λ) πc
1(Λ))
ails o be dense in R. We hen pick a poin 0∈Rsuch ha an open in e al Ja ound i has emp y
in e sec ion wi h
πa
1(Λ)∪(πb
1(Λ) πc
1(Λ)).
Bu hen πc
1(Λ)∩Jis dense in J, and χis locally he Fou ie ans o m o an L1(R) unc ion a ound
0. We hus ind a unc ion χ1which coincides wi h χon some open in e al I⊂Jwi h 0∈I,
while χ1is he Fou ie ans o m o an L1(R) unc ion. Nex , we pick g∈L1(R) wi h b
g( 0),0, such
ha suppb
g⋐I, and de ine ∈L1(R) ia b
=−χ1b
g, so ha (1.3) holds. This gi es us a non i ial
measu e µwi h he equi ed p ope ies, and so (Γ,Λ) canno be a Heisenbe g uniqueness pai . 
The c oss. I Γis a c oss, he PDE (1.1) exp esses he wa e equa ion. By he in a iance p ope ies
(in -1) and (in -2), we may es ic ou a en ion o he case when
Γ = (R×{0})∪({0}×R)
is he union o he wo axes. He e, i appea s ha he cha ac e iza ion o uniqueness pai s (Γ,Λ)
may ge qui e complica ed. Ob iously, i is a necessa y condi ion ha π1(Λ) and π2(Λ) be dense
(π2(Λ) is he o hogonal p ojec ion o he ξ2-axis). This is a om su icien , because i Λis
con ained in a smoo h g aph, we may un in o ouble. Fo ins ance, i Λis con ained in he
diagonal ξ1=ξ2, hen we may choose
dµ(x1,x2)= (x1) dx1dδ0(x2)− (x2) dx2dδ0(x1),
4 Hedenmalm and Mon es
whe e ∈L1(R), which is suppo ed on Γand non i ial gene ically, whileb
µ(ξ1, ξ2)=0 o ξ1=ξ2.
The ellipse. I Γis an ellipse, he in a iance o (in -1) and (in -2) allows us o ocus on he ci cle
Γ = {x=(x1,x2)∈R2:x2
1+x2
2=1}.
The co esponding PDE (1.1) is he eigen alue equa ion o he Laplacian. He e, he ac ha Γis
compac en ails ha b
µ(ξ) ex ends o an en i e unc ion o exponen ial ype in C2. I would seem
ha easonable c i e ia on Λmay be ound ha a e a leas close o being necessa y and su icien
o (Γ,Λ) o be a Heisenbe g uniqueness pai .
The pa abola. I Γis a pa abola, he in a iance o (in -1) and (in -2) allows us o ocus on he
pa abola
Γ = {x=(x1,x2)∈R2:x2=x2
1}.
The co esponding PDE (1.1) is he one-dimensional Sch ödinge equa ion wi hou po en ial. He e,
he p oblem o cha ac e izing he Heisenbe g uniqueness pai s (Γ,Λ) appea s qui e challenging.
The hype bola. We shall ocus mos o ou a en ion o he case when Γis a hype bola. The
co esponding PDE (1.1) is he one-dimensional Klein-Go don equa ion. We will see ha he
si ua ion wi h Heisenbe g uniqueness pai s is d ama ically di e en om ha o he c oss. By he
in a iance (in -1) and (in -2), we may assume ha he hype bola is gi en by
x1x2=1.
Theo em 1.2. Suppose Γis he hype bola x1x2=1and ha Λis he la ice-c oss
Λ = (αZ×{0})∪({0}×βZ),
whe e α, β a e posi i e eals. Then (Γ,Λ)is a Heisenbe g uniqueness pai i and only i αβ ≤1.
The emainde o his wo k is de o ed o p o ing his asse ion. Bu be o e we u n o he
p oo , le us conside a gene aliza ion which is mo e o less immedia e.
Co olla y 1.3. Suppose Γεis he hype bola x1x2=ε, whe e ε,0is eal, and ha Λis he la ice-c oss
Λ = (αZ×{0})∪({0}×βZ),
whe e α, β a e posi i e eals. Then (Γε,Λ)is a Heisenbe g uniqueness pai i and only i αβ ≤1/|ε|.
The eccen ici y o he hype bola Γεis √2 independen ly o ε. The condi ion o he co olla y
(αβ ≤1/|ε|) ge s weake as |ε|dec eases. Howe e , in he limi si ua ion ε=0 – he c oss – he
si ua ion changes d ama ically: i Λis con ained in he dual c oss (R×{0})∪({0}×R), hen Λmus
ac ually be dense in he c oss o (Γ0,Λ) o be a Heisenbe g uniqueness pai .
Rema k 1.4.Conside o a momen he se s
Λ′=([θ, +∞[×]−∞,0]) ∪(] −∞,0] ×[0,+∞[)
and
Λ′′ =n(ξ1, ξ2)∈R2:a1ξ1+a2ξ2=0o,
whe e θ, a1,a2a e all eal pa ame e s, subjec o θ > 0 and a1a2>0. The se Λ′is a guably mo e
massi e han he la ice-c oss Λo Co olla y 1.3. Ne e heless, i Γεis as in Co olla y 1.3, wi h
εposi i e, i can be shown ha (Γε,Λ′) ails o be a Heisenbe g uniqueness pai , no ma e wha
posi i e alues εand θassume. Analogously, (Γε,Λ′′) also ails o be a Heisenbe g uniqueness
pai , o all ε > 0 and a1a2>0 (bu i can be shown ha (Γε,Λ′∪Λ′′) is a Heisenbe g uniqueness
pai , howe e ). This sugges s ha i is c ucial ha he poin s o he la ice-c oss Λo Co olla y 1.3
a e loca ed along he cha ac e is ic di ec ions o he Klein-Go don equa ion ( he wo axes).
We need a esul o algeb aic na u e.
Heisenbe g uniqueness pai s and he Klein-Go don equa ion 5
Lemma 1.5. Le z1,z2∈Cbe wo poin s such ha
z1−z2=am ∈aZ,1
z1−1
z2
=bn ∈bZ,
o some posi i e eals a,b. Then, unless z1=z2, we ha e
z1=am
21± 1−4
abmn,z2=z1−am.
The p oo is a simple exe cise, and he e o e omi ed.
Rema k 1.6.Le us conside he singula measu e µ=δu−δ , whe e
u=(u1,1/u1)∈Γ, =( 1,1/ 1)∈Γ.
Then
b
µ(ξ)=eπi(ξ1u1+ξ2/u1)−eπi(ξ1 1+ξ2/ 1),
so ha
b
µ(ξ1,0) =eπiξ1u1−eπiξ1 1,b
µ(0, ξ2)=eπiξ2/u1−eπiξ2/ 1.
Suppose we y o achie e ha
(1.5) b
µ(αj,0) =b
µ(0, βk)=0,j,k∈Z,
o some posi i e eals α, β. We see ha his amoun s o
eπiαu1=eπiα 1,eπiβ/u1=eπiβ/ 1,
which we ew i e in he o m
u1− 1∈2
αZ,1
u1−1
1∈2
βZ.
In iew o Lemma 1.5, he e a e plen y o such poin s u1, 1∈Rwi h u1, 1, o any gi en α, β.
This shows ha he equi emen ha he measu e µbe absolu ely con inuous wi h espec o a c
leng h measu e on Γis essen ial; wi hou i , Theo em 1.2 would simply no be ue.
2. Dynamics o a Gauss- ype map
A Gauss- ype map. In o de o p o e ou main heo em (Theo em 1.2), we will need o s udy he
in a ian measu es o a pa icula map. We shall conside a map on he in e al ] −1,1], which
we hink o as R/2Z( opologically as well). The map in ques ion is de ined by U(0) =0 and
U(x)=−1
x2
,x,0,
whe e o eal , he exp ession { }2∈]−1,1] is he unique numbe such ha − { }2∈2Z. The
unc ion Uis locally s ic ly inc easing and con inuous, excep o being in e up ed by jumps.
The map U:] −1,1] →]−1,1] is associa ed wi h con inued ac ions wi h e en pa ial quo ien s (see
[10], [11], [7], [3]). We see ha , o j=±1,±2,±3,...,
U(x)=−1
x+2j,1
2j+1<x≤1
2j−1,
and hence Umaps he in e al ] 1
2j+1,1
2j−1] on o ] −1,1] in a one- o-one ashion. The de i a i e o
Uis locally
U′(x)=1
x2,x∈]−1,1] 1
2Z+1.
The poin 1 is a ixed poin o U, and U′(1−)=U′(−1+)=1, which makes 1 is a weakly epelling
ixed poin . This means ha when we i e a e U, once we a e close o 1 (which is he same poin as
−1 in R/2Z), he successi e i e a es will emain nea 1 o a long ime. I x∈]−1,1] is a ional,

6 Hedenmalm and Mon es
hen a e a ini e numbe o s eps, he U-i e a e o xis ei he 0 o 1 (see, o ins ance [7]). This
illumina es why i a ional numbe s end o spend a la ge po ion o hei U-o bi s nea 1.
In a ian measu es. I ϕis a con inuous unc ion on R/2Zand νis a bounded complex Bo el
measu e on ] −1,1], hen he in eg al
(2.1) Z]−1,1]
ϕ(x) dν(x)
is well-de ined. Howe e , he in eg al (2.1) makes sense unde weake assump ions on ϕ. Suppose
Eis an open subse o ] −1,1] such ha he complemen ] −1,1] Eis coun able, and ha ϕis
bounded on ] −1,1] and con inuous on E. Then (2.1) makes sense o ϕ, and we call he unc ion
ϕpseudo-con inuous. We ecall he amilia no ion ha a bounded complex Bo el measu e νon
]−1,1] is U-in a ian p o ided ha
(2.2) Z]−1,1]
ϕ(U(x)) dν(x)=Z]−1,1]
ϕ(x) dν(x)
holds o all pseudo-con inuous es unc ions ϕ; i is easy o see ha ϕ◦Uis pseudo-con inuous
i ϕis pseudo-con inuous, so ha (2.2) makes sense. We shall e o mula e his c i e ion in mo e
conc e e e ms. Fi s , we no e ha
Z]−1,1] {0}
ϕ(U(x)) dν(x)=X
j∈Z∗Z]1
2j+1,1
2j−1]
ϕ(U(x)) dν(x)=X
j∈Z∗Z]1
2j+1,1
2j−1]
ϕ−1
x+2jdν(x),
whe e Z∗=Z {0}, and ha
Z]1
2j+1,1
2j−1]
ϕ−1
x+2jdν(x)=Z]−1,1]
ϕ( ) dνj( ),
whe e
(2.3) dνj( )=dν1
2j− ,−1< ≤1,
so ha we ha e Z]−1,1] {0}
ϕ(U(x)) dν(x)=X
j∈Z∗Z]−1,1]
ϕ( ) dνj( ).
I ollows ha νis U-in a ian i and only i
(2.4) ν=ν({0})δ0+X
j∈Z∗
νj.
Mo e gene ally, gi en λ∈C, we wan o alk abou (U, λ)-in a ian measu es, de ined by he
equi emen ha
(2.5) Z]−1,1]
ϕ(U(x)) dν(x)=λZ]−1,1]
ϕ(x) dν(x)
hold o all es unc ions ϕ; speci ically, his means ha
(2.6) λν =ν({0})δ0+X
j∈Z∗
νj.
I is easy o see ha o |λ|>1, he e a e no (U, λ)-in a ian measu es excep o he ze o measu e.
P oposi ion 2.1. Suppose νis a bounded (U, λ)-in a ian measu e on ]−1,1], and w i e ν=νa+νs,
whe e νais absolu ely con inuous, while νsis singula . Then νaand νsa e also (U, λ)-in a ian . Mo eo e ,
i |λ|=1, hen |ν|,|νa|, and |νs|a e all U-in a ian measu es.
Heisenbe g uniqueness pai s and he Klein-Go don equa ion 7
P oo . The ela ion (2.6) spli s:
(2.7) νa=λX
j∈Z∗
(νj)a, νs=λν({0})δ0+X
j∈Z∗
(νj)s,
whe e he subsc ip s aand sindica e he absolu ely con inuous and singula pa s, espec i ely, o
he measu e in ques ion. We easily ealize ha (νj)a=(νa)jand (νj)s=(νs)j, so ha (2.7) exp esses
ha νaand νsa e bo h U-in a ian .
Nex , we suppose |λ|=1, and u n o he asse ion ha |ν|is U-in a ian . Taking absolu e
alues, we ha e
(2.8) |dν( )| ≤ |ν({0})|dδ0( )+X
j∈Z∗|dνj( )|,
and so
Z]−1,1] |dν( )|≤ X
j∈Z∗Z]−1,1] |dνj( )|=|ν({0})|+X
j∈Z∗Z]1
2j+1,1
2j−1]|dν( )|
=|ν({0})|+Z]−1,1] {0}|dν( )|=Z]−1,1] |dν( )|.
This is only possible i we ha e in ac equali y in (2.8):
|dν( )|=|ν({0})|dδ0( )+X
j∈Z∗|dνj( )|.
This ela ion exp esses ha |ν|is U-in a ian ; ha |νa|and |νs|a e U-in a ian is a simple conse-
quence o his ac . 
An unbounded smoo h in a ian measu e. We now conside he posi i e unbounded smoo h
measu e
dω(x)=dx
1−x2.
The c i e ion (2.6) makes sense al hough ωis unbounded. The ollowing asse ion was essen ially
ound by Schweige [10].
P oposi ion 2.2. The measu e ωis U-in a ian .
We supply he simple p oo .
P oo . We check ha
dωj( )=dω1
2j− =d
(2j− )2−1,
and since
X
j∈Z∗
1
(2j− )2−1=1
2X
j∈Z∗1
2j− −1−1
2j− +1=1
21
1+ +1
1− =1
1− 2,
we ind om (2.6) ha ωis U-in a ian . 
Schweige [10] ac ually ocused on he ela ed map |U|: [0,1] →[0,1] gi en by |U|(x)=|U(x)|.
He ob ained he ollowing basic esul .
P oposi ion 2.3. The measu e ωis in a ian also wi h espec o |U|. Mo eo e , |U|is e godic, ha is, i
E⊂[0,1] is a |U|-in a ian se , hen ei he ω(E)=0o ω([0,1] E)=0.
8 Hedenmalm and Mon es
Consequences o E godic Theo y. The Bi kho E godic Theo em – in his se ing o an unbounded
in a ian e godic measu e [1] – s a es ha i ϕis Bo el measu able and e en wi h
Z1
−1
|ϕ( )|
1− 2d <+∞,
hen
1
N
N−1
X
k=0
ϕ(Uhki( )) →0 as N→+∞
almos e e ywhe e on ] −1,1]. He e, Uhkis ands o he k- h i e a e o U. We obse e ha we do
no need o know whe he Uis e godic, jus ha |U|is, i we use ha |U(−x)|=|U(x)|. We pick
ϕ( )=1− 2, and ge :
(2.9) 1
N
N−1
X
k=01−|Uhki( )|2→0 as N→+∞
almos e e ywhe e on ]−1,1]. Suppose νis a posi i e, bounded, and absolu ely con inuous U-in a ian
measu e on ] −1,1]. By he U-in a iance, we ha e
(2.10) Z]−1,1] 1−|Uhki( )|2dν( )=Z]−1,1]
(1 − 2) dν( ),
and so Z]−1,1]
1
N
N−1
X
k=01−|Uhki( )|2dν( )=Z]−1,1]
(1 − 2) dν( ).
By he Lebesgue domina ed con e gence heo em, i ollows om (2.9) ha
Z]−1,1]
1
N
N−1
X
k=01−|Uhki( )|2dν( )→0,as N→+∞,
which combined wi h he (2.10) leads o
Z]−1,1]
(1 − 2) dν( )=0.
This is only possible i ν=0.
We o malize his in a p oposi ion.
P oposi ion 2.4. Suppose λ∈Chas |λ|=1, and ha νis an absolu ely con inuous bounded complex
(U, λ)-in a ian Bo el measu e on ]−1,1]. Then ν=0.
P oo . By P oposi ion 2.1, |ν|is a U-in a ian measu e. By he abo e a gumen , |ν|=0, and so
ν=0. 
3. Ex ension o he igonome ic sys em
The igonome ic sys em. The igonome ic sys em {en(x)}n∈Z, wi h en(x)=eπinx, is e y suc-
cess ul in desc ibing 2-pe iodic unc ions on he line. Ha ald Boh – he b o he o Niels Boh ,
he physicis – de eloped o e a numbe o yea s in he 1920s and 1930s he heo y o almos
pe iodic unc ions based on mo e gene al eal equencies a he han he in ege equencies o
he igonome ic sys em.
An ex ension o he igonome ic sys em. He e, we conside ano he ex ension o he igonome -
ic sys em, connec ed wi h he heo y o composi ion ope a o s. Le βbe a posi i e eal pa ame e .
We in oduce, o in ege s n,
ehβi
n(x)=enβ
x=eπiβn/x,
Heisenbe g uniqueness pai s and he Klein-Go don equa ion 9
and no e ha hese unc ions a e bounded on he eal line.
A e a dila ion o he line, Theo em 1.2 is equi alen o he ollowing s a emen .
Theo em 3.1. As n anges o e he in ege s, he unc ions en(x)and ehβi
n(x) o m a weak-s a -spanning
sys em in L∞(R)i and only i 0< β ≤1.
I µis a posi i e bounded absolu ely con inuous Bo el measu e on R, hen a bounded unc ion
in L∞(R) is au oma ically in Lp(R, µ) o 1 <p<+∞, and he weak-s a closu e o a subspace in
L∞(R) is con ained in he no m closu e in Lp(R, µ). We hen ha e he ollowing consequence o
Theo em 3.1. The necessi y pa jus equi es mimicking he co esponding a gumen in ol ing
ha monic ex ensions in Sec ion 4 below.
Co olla y 3.2. Suppose 1<p<+∞, and ha dµ(x)=M(x)dx, whe e M(x)≥0is Bo el measu able,
wi h
0<Z+∞
−∞
M(x) dx<+∞.
Then, as n anges o e he in ege s, he unc ions en(x)and ehβi
n(x) o m a spanning sys em in Lp(R, µ)
p o ided ha 0< β ≤1. I , in addi ion,
Z+∞
−∞
dx
(1 +x2)p/(p−1)M(x)1/(p−1) <+∞,
he condi ion 0< β ≤1is also necessa y in o de o ha e a spanning sys em.
4. Necessi y o he condi ion 0< β ≤1
Ha monic ex ension. We ex end he unc ions enha monically and boundedly o he uppe hal
plane C+={z∈C: Im z>0}:
en(z)=eπinz,Im z≥0,n≥0,
while
en(z)=eπin¯
z,Im z≥0,n<0.
Likewise, he ha monic ex ension o ehβi
nis
ehβi
n(z)=eπiβn/¯
z,Im z≥0,n≥0,
and
ehβi
n(z)=eπiβn/z,Im z≥0,n<0.
Poin sepa a ion. A gene al L∞(R) unc ion is ex ended ha monically and boundedly o C+ ia
he Poisson ke nel; o each z0=x0+iy0∈C+, he poin e alua ion unc ional 7→ (z0) is gi en
by
(z0)=1
πZ+∞
−∞
P( ,z0) ( ) d ,P( ,z0)=y0
(x0− )2+y2
0
,
whe e 7→ P( ,z0) is in L1(R), and he unc ional is he e o e weak-s a con inuous on L∞(R). As
we ha monically ex end all he unc ions in L∞(R), we ge he space o all bounded ha monic
unc ions in C+. The bounded ha monic unc ions in C+sepa a e he poin s o C+, so i we can
ind wo poin s z1,z2∈C+wi h z1,z2, such ha
(4.1) en(z1)=en(z2),ehβi
n(z1)=ehβi
n(z2),
o all n∈Z, hen he linea span o en,ehβi
n, canno be weak-s a dense in L∞(R). The condi ion
(4.1) boils down o
z1−z2∈2Z,1
z1−1
z2∈2
βZ,
16 Hedenmalm and Mon es
Hedenmalm: Depa men o Ma hema ics, The Royal Ins i u e o Technology, S – 100 44 S ockholm,
SWEDEN
E-mail add ess:haakanh@ma h.k h.se
Mon es-Rod íguez: Depa men o Ma hema ical Analysis, Uni e si y o Se illa, Se illa, SPAIN
E-mail add ess:amon [email protected]